Suppose that a person has a lottery ticket from which she will win $X$ dollars, where $X \sim\mathrm{ Unif} (0,4)$. Suppose her utility function is $U(x) = x\alpha$ for $x \geq 0$ and $0$ otherwise, ...

Let $U \sim \mathrm{Unif}(0,1)$, $X=U^2$ and $Y=e^X$. Compute $E[Y]$ (leave answer as an integral).
So essentially we need to compute $E[e^{U^2}]$? I am a little confused how to approach this problem?
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Came across a problem that I worked on sometime ago having the following structure:
Given an opaque container (or locomotive with so many passenger cars, etc) that has--with equal probability--1 to N ...

I have a Perlin noise algorithm I've written my self. It seems to produce gausian numbers at the range of -1.5 and 1.5 but I'll convert them to the range of -1 and 1. I' currently working on a project ...

Why PDF of $g(X)=X^3$ is not uniformly distributed, when X is uniform random variable between $(0,1)$? As for every value of X there is unique value of $g(X)$, hence the probability density of $g(X)$ ...

If $N$ takes the values $0, 1$ and $2$ with probabilities $½, ¼ $ and $¼ $ respectively, and the $X_i$ ’s have a $U(0,10)$ distribution, draw a sketch of the frequency distribution of $S$.
$N$ is the ...

I'm chasing a bug in the RNG for a well-known programming language under certain pathological inputs. There is an obvious pattern in this pathological case, apparent with very small n (~ 10000), and ...

I have a probability $P$ derived from:
- A random integer $A$ uniformly distributed on its range such that $A\in\left[0, 100\right]$
- An integer $K$ such that $K\in\Bbb N$
- A number $X$ such that ...

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong?
Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...

Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out?
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Let $W$ be the occurence meaning the following ordering : $X_1...X_k$ where $X_k$ is greatest.. $X_k$ is greatest, and next in order is $X_1$, and the order of the others is not important. Because of ...

In a exercise i'm doing it is asked to find the maximum likelihood estimator of a random sample $X_{1}, ... , X_{n}$ of a population with distribution $X\sim U(- \theta , \theta) $. I've found that ...

In a simple setting, $w$ is uniformly distributed on $[0,1]$, R is a function of $wd$.
I want to find optimal d in this expression,
$aR-(d^2-1)/2$.
When I try to find out optimal $d$ than it is $0$. ...